which follows from [1]}.
Analogously, \(\operatorname{hoRan}_\iota ^{\prime } \colon {H}_\infty ^p \rightarrow \widetilde{{H}}_\infty ^p\)  is a right Quillen functor (since \(Q^{\prime }\)  is homotopical and valued in projectively cofibrant simplicial presheaves).
Further, the natural transformation \(\eta \colon \operatorname{hoRan}_\iota \longrightarrow \operatorname{hoRan}_\iota ^{\prime }\)  establishes that \(\operatorname{hoRan}_\iota ^{\prime }\)  maps local objects in \({H}_\infty ^{p \ell }\)  to local objects in \(\widetilde{{H}}_\infty ^{p \ell }\) , and hence that we also have a Quillen adjunction
\(\begin{tikzcd}Q^{\prime } \circ \iota ^* : \widetilde{{H}}_\infty ^{p \ell } [r, shift left=0.14cm, "\perp "^{\prime } yshift=0.05cm]& {H}_\infty ^{p \ell } : \operatorname{hoRan}_\iota ^{\prime }\,. [l, shift left=0.14cm]\end{tikzcd}\) 
